#7758028

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4733 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

(9,10,13,12)(11,15,16,14)(17,18,22,23)(19,25,28,24)(20,21,27,26)(29,30,32,31);(1,2)(3,8)(4,7)(5,6)(9,11,13,16)(10,14,12,15)(18,23)(19,28)(20,27)(30,31);(2,6)(7,8)(9,10,13,12)(11,14,16,15)(17,19,22,28)(18,24,23,25)(20,29,27,32)(21,31,26,30);(1,3,5,4)(2,7,6,8)(9,10,13,12)(11,14,16,15)(17,19)(18,25)(20,29)(21,30)(22,28)(23,24)(26,31)(27,32);(1,4,5,3)(2,8,6,7)(17,20,22,27)(18,26,23,21)(19,29,28,32)(24,30,25,31);(1,2)(3,7)(4,8)(5,6)(9,12,13,10)(11,15,16,14)(17,21,22,26)(18,27,23,20)(19,30,28,31)(24,29,25,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7665224);